Irrationality of the Zeta Constants

A general technique for proving the irrationality of the zeta constants $\zeta(s)$ for odd $s = 2n + 1 \geq 3$ from the known irrationality of the beta constants $L(2n+1)$ is developed in this note. The results on the irrationality of the zeta constants $\zeta(2n)$, where $n\geq 1$, and $\zeta(3)$ are well known, but the results on the irrationality for the zeta constants $\zeta(2n+1)$, where $n \geq 2$, are new, and these results seem to confirm that these constants are irrational numbers.

💡 Research Summary

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The paper under review claims to prove that every odd zeta value (\zeta(s)) with (s=2n+1\ge 3) is irrational, by exploiting the already‑known irrationality (indeed transcendence) of the corresponding Dirichlet beta‑type L‑values (L(2n+1,\chi)). The author’s strategy is to use the factorisation of the Dedekind zeta function of a quadratic field, \